LMFDB - Embedded newform 1148.4.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.7341926866$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 247 x^{13} - 6 x^{12} + 23870 x^{11} + 940 x^{10} - 1147074 x^{9} - 8966 x^{8} + \cdots + 1720288256 $ x^15 - 247x^13 - 6x^12 + 23870x^11 + 940x^10 - 1147074x^9 - 8966x^8 + 28798759x^7 - 3139244x^6 - 363586916x^5 + 114207044x^4 + 2060613491x^3 - 892545712x^2 - 4098298668*x + 1720288256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.99072$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.99072 q^{3} +8.14355 q^{5} -7.00000 q^{7} -23.0370 q^{9} +O(q^{10})$	$q-1.99072 q^{3} +8.14355 q^{5} -7.00000 q^{7} -23.0370 q^{9} +10.9570 q^{11} -37.6085 q^{13} -16.2115 q^{15} +85.7382 q^{17} +84.4884 q^{19} +13.9350 q^{21} +20.4238 q^{23} -58.6825 q^{25} +99.6096 q^{27} +84.5046 q^{29} -247.587 q^{31} -21.8122 q^{33} -57.0049 q^{35} +65.8557 q^{37} +74.8679 q^{39} -41.0000 q^{41} +66.5805 q^{43} -187.603 q^{45} -209.791 q^{47} +49.0000 q^{49} -170.681 q^{51} -423.543 q^{53} +89.2286 q^{55} -168.193 q^{57} -570.165 q^{59} -386.821 q^{61} +161.259 q^{63} -306.267 q^{65} +1082.75 q^{67} -40.6580 q^{69} -887.746 q^{71} +666.481 q^{73} +116.820 q^{75} -76.6987 q^{77} -75.7614 q^{79} +423.706 q^{81} +251.401 q^{83} +698.214 q^{85} -168.225 q^{87} -584.785 q^{89} +263.260 q^{91} +492.875 q^{93} +688.036 q^{95} -733.043 q^{97} -252.416 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) $ 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9 - 20 * q^11 - 70 * q^13 - 20 * q^15 + 160 * q^17 - 6 * q^19 - 118 * q^23 + 569 * q^25 + 18 * q^27 - 162 * q^29 - 164 * q^31 - 292 * q^33 - 42 * q^35 - 410 * q^37 - 206 * q^39 - 615 * q^41 - 1022 * q^43 + 196 * q^45 - 628 * q^47 + 735 * q^49 - 1994 * q^51 - 512 * q^53 - 1128 * q^55 - 266 * q^57 - 144 * q^59 - 256 * q^61 - 623 * q^63 - 1000 * q^65 - 2670 * q^67 + 108 * q^69 - 1048 * q^71 - 606 * q^73 - 3796 * q^75 + 140 * q^77 - 1386 * q^79 - 2541 * q^81 - 2022 * q^83 - 2848 * q^85 - 3700 * q^87 - 500 * q^89 + 490 * q^91 - 2194 * q^93 - 5230 * q^95 + 1326 * q^97 - 2732 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.99072	−0.383114	−0.191557	−	0.981482i	$-0.561354\pi$
$3$	−1.99072	−0.383114	−0.191557	+	0.981482i	$0.561354\pi$
$4$	0	0
$5$	8.14355	0.728382	0.364191	−	0.931324i	$-0.381346\pi$
$5$	8.14355	0.728382	0.364191	+	0.931324i	$0.381346\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−23.0370	−0.853224
$10$	0	0
$11$	10.9570	0.300332	0.150166	−	0.988661i	$-0.452019\pi$
$11$	10.9570	0.300332	0.150166	+	0.988661i	$0.452019\pi$
$12$	0	0
$13$	−37.6085	−0.802364	−0.401182	−	0.915998i	$-0.631400\pi$
$13$	−37.6085	−0.802364	−0.401182	+	0.915998i	$0.631400\pi$
$14$	0	0
$15$	−16.2115	−0.279053
$16$	0	0
$17$	85.7382	1.22321	0.611605	−	0.791163i	$-0.290524\pi$
$17$	85.7382	1.22321	0.611605	+	0.791163i	$0.290524\pi$
$18$	0	0
$19$	84.4884	1.02016	0.510078	−	0.860128i	$-0.329616\pi$
$19$	84.4884	1.02016	0.510078	+	0.860128i	$0.329616\pi$
$20$	0	0
$21$	13.9350	0.144803
$22$	0	0
$23$	20.4238	0.185159	0.0925795	−	0.995705i	$-0.470489\pi$
$23$	20.4238	0.185159	0.0925795	+	0.995705i	$0.470489\pi$
$24$	0	0
$25$	−58.6825	−0.469460
$26$	0	0
$27$	99.6096	0.709996
$28$	0	0
$29$	84.5046	0.541107	0.270554	−	0.962705i	$-0.412793\pi$
$29$	84.5046	0.541107	0.270554	+	0.962705i	$0.412793\pi$
$30$	0	0
$31$	−247.587	−1.43445	−0.717224	−	0.696843i	$-0.754587\pi$
$31$	−247.587	−1.43445	−0.717224	+	0.696843i	$0.754587\pi$
$32$	0	0
$33$	−21.8122	−0.115061
$34$	0	0
$35$	−57.0049	−0.275302
$36$	0	0
$37$	65.8557	0.292611	0.146306	−	0.989239i	$-0.453262\pi$
$37$	65.8557	0.292611	0.146306	+	0.989239i	$0.453262\pi$
$38$	0	0
$39$	74.8679	0.307397
$40$	0	0
$41$	−41.0000	−0.156174
$41$	−41.0000	−0.156174
$42$	0	0
$43$	66.5805	0.236126	0.118063	−	0.993006i	$-0.462331\pi$
$43$	66.5805	0.236126	0.118063	+	0.993006i	$0.462331\pi$
$44$	0	0
$45$	−187.603	−0.621472
$46$	0	0
$47$	−209.791	−0.651089	−0.325545	−	0.945527i	$-0.605548\pi$
$47$	−209.791	−0.651089	−0.325545	+	0.945527i	$0.605548\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−170.681	−0.468629
$52$	0	0
$53$	−423.543	−1.09770	−0.548850	−	0.835921i	$-0.684934\pi$
$53$	−423.543	−1.09770	−0.548850	+	0.835921i	$0.684934\pi$
$54$	0	0
$55$	89.2286	0.218756
$56$	0	0
$57$	−168.193	−0.390836
$58$	0	0
$59$	−570.165	−1.25812	−0.629061	−	0.777356i	$-0.716560\pi$
$59$	−570.165	−1.25812	−0.629061	+	0.777356i	$0.716560\pi$
$60$	0	0
$61$	−386.821	−0.811924	−0.405962	−	0.913890i	$-0.633064\pi$
$61$	−386.821	−0.811924	−0.405962	+	0.913890i	$0.633064\pi$
$62$	0	0
$63$	161.259	0.322488
$64$	0	0
$65$	−306.267	−0.584427
$66$	0	0
$67$	1082.75	1.97432	0.987159	−	0.159738i	$-0.0510650\pi$
$67$	1082.75	1.97432	0.987159	+	0.159738i	$0.0510650\pi$
$68$	0	0
$69$	−40.6580	−0.0709370
$70$	0	0
$71$	−887.746	−1.48389	−0.741944	−	0.670462i	$-0.766096\pi$
$71$	−887.746	−1.48389	−0.741944	+	0.670462i	$0.766096\pi$
$72$	0	0
$73$	666.481	1.06857	0.534285	−	0.845304i	$-0.320581\pi$
$73$	666.481	1.06857	0.534285	+	0.845304i	$0.320581\pi$
$74$	0	0
$75$	116.820	0.179857
$76$	0	0
$77$	−76.6987	−0.113515
$78$	0	0
$79$	−75.7614	−0.107897	−0.0539483	−	0.998544i	$-0.517181\pi$
$79$	−75.7614	−0.107897	−0.0539483	+	0.998544i	$0.517181\pi$
$80$	0	0
$81$	423.706	0.581215
$82$	0	0
$83$	251.401	0.332468	0.166234	−	0.986086i	$-0.446839\pi$
$83$	251.401	0.332468	0.166234	+	0.986086i	$0.446839\pi$
$84$	0	0
$85$	698.214	0.890964
$86$	0	0
$87$	−168.225	−0.207306
$88$	0	0
$89$	−584.785	−0.696484	−0.348242	−	0.937405i	$-0.613221\pi$
$89$	−584.785	−0.696484	−0.348242	+	0.937405i	$0.613221\pi$
$90$	0	0
$91$	263.260	0.303265
$92$	0	0
$93$	492.875	0.549557
$94$	0	0
$95$	688.036	0.743063
$96$	0	0
$97$	−733.043	−0.767312	−0.383656	−	0.923476i	$-0.625335\pi$
$97$	−733.043	−0.767312	−0.383656	+	0.923476i	$0.625335\pi$
$98$	0	0
$99$	−252.416	−0.256250
$100$	0	0
$101$	−667.210	−0.657325	−0.328663	−	0.944447i	$-0.606598\pi$
$101$	−667.210	−0.657325	−0.328663	+	0.944447i	$0.606598\pi$
$102$	0	0
$103$	−1104.03	−1.05615	−0.528074	−	0.849198i	$-0.677086\pi$
$103$	−1104.03	−1.05615	−0.528074	+	0.849198i	$0.677086\pi$
$104$	0	0
$105$	113.481	0.105472
$106$	0	0
$107$	−1257.09	−1.13577	−0.567887	−	0.823107i	$-0.692239\pi$
$107$	−1257.09	−1.13577	−0.567887	+	0.823107i	$0.692239\pi$
$108$	0	0
$109$	1074.00	0.943770	0.471885	−	0.881660i	$-0.343574\pi$
$109$	1074.00	0.943770	0.471885	+	0.881660i	$0.343574\pi$
$110$	0	0
$111$	−131.100	−0.112103
$112$	0	0
$113$	1361.65	1.13357	0.566786	−	0.823865i	$-0.308187\pi$
$113$	1361.65	1.13357	0.566786	+	0.823865i	$0.308187\pi$
$114$	0	0
$115$	166.322	0.134866
$116$	0	0
$117$	866.389	0.684596
$118$	0	0
$119$	−600.168	−0.462330
$120$	0	0
$121$	−1210.94	−0.909801
$122$	0	0
$123$	81.6194	0.0598323
$124$	0	0
$125$	−1495.83	−1.07033
$126$	0	0
$127$	−2573.70	−1.79826	−0.899129	−	0.437684i	$-0.855799\pi$
$127$	−2573.70	−1.79826	−0.899129	+	0.437684i	$0.855799\pi$
$128$	0	0
$129$	−132.543	−0.0904632
$130$	0	0
$131$	−1604.46	−1.07010	−0.535048	−	0.844822i	$-0.679706\pi$
$131$	−1604.46	−1.07010	−0.535048	+	0.844822i	$0.679706\pi$
$132$	0	0
$133$	−591.419	−0.385583
$134$	0	0
$135$	811.176	0.517148
$136$	0	0
$137$	1072.48	0.668821	0.334410	−	0.942428i	$-0.391463\pi$
$137$	1072.48	0.668821	0.334410	+	0.942428i	$0.391463\pi$
$138$	0	0
$139$	2517.27	1.53606	0.768029	−	0.640415i	$-0.221238\pi$
$139$	2517.27	1.53606	0.768029	+	0.640415i	$0.221238\pi$
$140$	0	0
$141$	417.635	0.249441
$142$	0	0
$143$	−412.075	−0.240975
$144$	0	0
$145$	688.167	0.394132
$146$	0	0
$147$	−97.5452	−0.0547305
$148$	0	0
$149$	352.490	0.193806	0.0969031	−	0.995294i	$-0.469106\pi$
$149$	352.490	0.193806	0.0969031	+	0.995294i	$0.469106\pi$
$150$	0	0
$151$	−2631.45	−1.41817	−0.709087	−	0.705121i	$-0.750893\pi$
$151$	−2631.45	−1.41817	−0.709087	+	0.705121i	$0.750893\pi$
$152$	0	0
$153$	−1975.16	−1.04367
$154$	0	0
$155$	−2016.24	−1.04483
$156$	0	0
$157$	−2243.62	−1.14051	−0.570257	−	0.821466i	$-0.693156\pi$
$157$	−2243.62	−1.14051	−0.570257	+	0.821466i	$0.693156\pi$
$158$	0	0
$159$	843.155	0.420544
$160$	0	0
$161$	−142.967	−0.0699835
$162$	0	0
$163$	−2411.85	−1.15896	−0.579481	−	0.814986i	$-0.696745\pi$
$163$	−2411.85	−1.15896	−0.579481	+	0.814986i	$0.696745\pi$
$164$	0	0
$165$	−177.629	−0.0838085
$166$	0	0
$167$	−1745.39	−0.808756	−0.404378	−	0.914592i	$-0.632512\pi$
$167$	−1745.39	−0.808756	−0.404378	+	0.914592i	$0.632512\pi$
$168$	0	0
$169$	−782.599	−0.356213
$170$	0	0
$171$	−1946.36	−0.870422
$172$	0	0
$173$	2877.69	1.26466	0.632331	−	0.774698i	$-0.282098\pi$
$173$	2877.69	1.26466	0.632331	+	0.774698i	$0.282098\pi$
$174$	0	0
$175$	410.778	0.177439
$176$	0	0
$177$	1135.04	0.482004
$178$	0	0
$179$	1604.93	0.670159	0.335079	−	0.942190i	$-0.391237\pi$
$179$	1604.93	0.670159	0.335079	+	0.942190i	$0.391237\pi$
$180$	0	0
$181$	2650.92	1.08863	0.544313	−	0.838882i	$-0.316791\pi$
$181$	2650.92	1.08863	0.544313	+	0.838882i	$0.316791\pi$
$182$	0	0
$183$	770.052	0.311059
$184$	0	0
$185$	536.299	0.213132
$186$	0	0
$187$	939.431	0.367369
$188$	0	0
$189$	−697.267	−0.268353
$190$	0	0
$191$	−4404.98	−1.66876	−0.834381	−	0.551188i	$-0.814175\pi$
$191$	−4404.98	−1.66876	−0.834381	+	0.551188i	$0.814175\pi$
$192$	0	0
$193$	−4686.98	−1.74806	−0.874031	−	0.485870i	$-0.838503\pi$
$193$	−4686.98	−1.74806	−0.874031	+	0.485870i	$0.838503\pi$
$194$	0	0
$195$	609.691	0.223902
$196$	0	0
$197$	−4753.72	−1.71923	−0.859616	−	0.510941i	$-0.829297\pi$
$197$	−4753.72	−1.71923	−0.859616	+	0.510941i	$0.829297\pi$
$198$	0	0
$199$	−2961.03	−1.05478	−0.527391	−	0.849622i	$-0.676830\pi$
$199$	−2961.03	−1.05478	−0.527391	+	0.849622i	$0.676830\pi$
$200$	0	0
$201$	−2155.46	−0.756389
$202$	0	0
$203$	−591.532	−0.204519
$204$	0	0
$205$	−333.886	−0.113754
$206$	0	0
$207$	−470.504	−0.157982
$208$	0	0
$209$	925.737	0.306385
$210$	0	0
$211$	5187.43	1.69250	0.846250	−	0.532786i	$-0.178855\pi$
$211$	5187.43	1.69250	0.846250	+	0.532786i	$0.178855\pi$
$212$	0	0
$213$	1767.25	0.568498
$214$	0	0
$215$	542.202	0.171990
$216$	0	0
$217$	1733.11	0.542170
$218$	0	0
$219$	−1326.77	−0.409384
$220$	0	0
$221$	−3224.49	−0.981460
$222$	0	0
$223$	−4760.38	−1.42950	−0.714750	−	0.699380i	$-0.753459\pi$
$223$	−4760.38	−1.42950	−0.714750	+	0.699380i	$0.753459\pi$
$224$	0	0
$225$	1351.87	0.400555
$226$	0	0
$227$	−5386.00	−1.57481	−0.787403	−	0.616438i	$-0.788575\pi$
$227$	−5386.00	−1.57481	−0.787403	+	0.616438i	$0.788575\pi$
$228$	0	0
$229$	3407.13	0.983185	0.491592	−	0.870825i	$-0.336415\pi$
$229$	3407.13	0.983185	0.491592	+	0.870825i	$0.336415\pi$
$230$	0	0
$231$	152.686	0.0434890
$232$	0	0
$233$	2610.16	0.733893	0.366947	−	0.930242i	$-0.380403\pi$
$233$	2610.16	0.733893	0.366947	+	0.930242i	$0.380403\pi$
$234$	0	0
$235$	−1708.45	−0.474241
$236$	0	0
$237$	150.820	0.0413366
$238$	0	0
$239$	−3984.75	−1.07846	−0.539231	−	0.842158i	$-0.681285\pi$
$239$	−3984.75	−1.07846	−0.539231	+	0.842158i	$0.681285\pi$
$240$	0	0
$241$	762.391	0.203776	0.101888	−	0.994796i	$-0.467512\pi$
$241$	762.391	0.203776	0.101888	+	0.994796i	$0.467512\pi$
$242$	0	0
$243$	−3532.94	−0.932667
$244$	0	0
$245$	399.034	0.104055
$246$	0	0
$247$	−3177.49	−0.818537
$248$	0	0
$249$	−500.468	−0.127373
$250$	0	0
$251$	−5617.40	−1.41262	−0.706309	−	0.707904i	$-0.749641\pi$
$251$	−5617.40	−1.41262	−0.706309	+	0.707904i	$0.749641\pi$
$252$	0	0
$253$	223.783	0.0556091
$254$	0	0
$255$	−1389.95	−0.341341
$256$	0	0
$257$	1269.51	0.308132	0.154066	−	0.988061i	$-0.450763\pi$
$257$	1269.51	0.308132	0.154066	+	0.988061i	$0.450763\pi$
$258$	0	0
$259$	−460.990	−0.110597
$260$	0	0
$261$	−1946.74	−0.461685
$262$	0	0
$263$	3023.59	0.708906	0.354453	−	0.935074i	$-0.384667\pi$
$263$	3023.59	0.708906	0.354453	+	0.935074i	$0.384667\pi$
$264$	0	0
$265$	−3449.15	−0.799545
$266$	0	0
$267$	1164.14	0.266832
$268$	0	0
$269$	−7272.42	−1.64835	−0.824177	−	0.566333i	$-0.808362\pi$
$269$	−7272.42	−1.64835	−0.824177	+	0.566333i	$0.808362\pi$
$270$	0	0
$271$	7768.44	1.74133	0.870663	−	0.491880i	$-0.163690\pi$
$271$	7768.44	1.74133	0.870663	+	0.491880i	$0.163690\pi$
$272$	0	0
$273$	−524.076	−0.116185
$274$	0	0
$275$	−642.983	−0.140994
$276$	0	0
$277$	−1346.69	−0.292111	−0.146056	−	0.989276i	$-0.546658\pi$
$277$	−1346.69	−0.292111	−0.146056	+	0.989276i	$0.546658\pi$
$278$	0	0
$279$	5703.67	1.22391
$280$	0	0
$281$	8287.82	1.75947	0.879733	−	0.475469i	$-0.157721\pi$
$281$	8287.82	1.75947	0.879733	+	0.475469i	$0.157721\pi$
$282$	0	0
$283$	−447.572	−0.0940119	−0.0470060	−	0.998895i	$-0.514968\pi$
$283$	−447.572	−0.0940119	−0.0470060	+	0.998895i	$0.514968\pi$
$284$	0	0
$285$	−1369.69	−0.284678
$286$	0	0
$287$	287.000	0.0590281
$288$	0	0
$289$	2438.05	0.496244
$290$	0	0
$291$	1459.28	0.293968
$292$	0	0
$293$	7863.94	1.56797	0.783987	−	0.620778i	$-0.213183\pi$
$293$	7863.94	1.56797	0.783987	+	0.620778i	$0.213183\pi$
$294$	0	0
$295$	−4643.17	−0.916392
$296$	0	0
$297$	1091.42	0.213234
$298$	0	0
$299$	−768.109	−0.148565
$300$	0	0
$301$	−466.064	−0.0892474
$302$	0	0
$303$	1328.23	0.251830
$304$	0	0
$305$	−3150.10	−0.591391
$306$	0	0
$307$	2243.88	0.417150	0.208575	−	0.978006i	$-0.433118\pi$
$307$	2243.88	0.417150	0.208575	+	0.978006i	$0.433118\pi$
$308$	0	0
$309$	2197.81	0.404625
$310$	0	0
$311$	−794.483	−0.144858	−0.0724292	−	0.997374i	$-0.523075\pi$
$311$	−794.483	−0.144858	−0.0724292	+	0.997374i	$0.523075\pi$
$312$	0	0
$313$	9716.36	1.75464	0.877318	−	0.479910i	$-0.159331\pi$
$313$	9716.36	1.75464	0.877318	+	0.479910i	$0.159331\pi$
$314$	0	0
$315$	1313.22	0.234895
$316$	0	0
$317$	8565.78	1.51767	0.758837	−	0.651281i	$-0.225768\pi$
$317$	8565.78	1.51767	0.758837	+	0.651281i	$0.225768\pi$
$318$	0	0
$319$	925.913	0.162512
$320$	0	0
$321$	2502.52	0.435130
$322$	0	0
$323$	7243.89	1.24787
$324$	0	0
$325$	2206.96	0.376678
$326$	0	0
$327$	−2138.04	−0.361571
$328$	0	0
$329$	1468.54	0.246089
$330$	0	0
$331$	7265.48	1.20649	0.603243	−	0.797558i	$-0.293875\pi$
$331$	7265.48	1.20649	0.603243	+	0.797558i	$0.293875\pi$
$332$	0	0
$333$	−1517.12	−0.249663
$334$	0	0
$335$	8817.46	1.43806
$336$	0	0
$337$	−1100.25	−0.177847	−0.0889237	−	0.996038i	$-0.528343\pi$
$337$	−1100.25	−0.177847	−0.0889237	+	0.996038i	$0.528343\pi$
$338$	0	0
$339$	−2710.67	−0.434287
$340$	0	0
$341$	−2712.80	−0.430810
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−331.101	−0.0516692
$346$	0	0
$347$	−8966.20	−1.38712	−0.693560	−	0.720398i	$-0.743959\pi$
$347$	−8966.20	−1.38712	−0.693560	+	0.720398i	$0.743959\pi$
$348$	0	0
$349$	−5897.64	−0.904566	−0.452283	−	0.891874i	$-0.649390\pi$
$349$	−5897.64	−0.904566	−0.452283	+	0.891874i	$0.649390\pi$
$350$	0	0
$351$	−3746.17	−0.569675
$352$	0	0
$353$	−8840.11	−1.33289	−0.666447	−	0.745552i	$-0.732186\pi$
$353$	−8840.11	−1.33289	−0.666447	+	0.745552i	$0.732186\pi$
$354$	0	0
$355$	−7229.40	−1.08084
$356$	0	0
$357$	1194.76	0.177125
$358$	0	0
$359$	−1691.08	−0.248613	−0.124306	−	0.992244i	$-0.539671\pi$
$359$	−1691.08	−0.248613	−0.124306	+	0.992244i	$0.539671\pi$
$360$	0	0
$361$	279.298	0.0407199
$362$	0	0
$363$	2410.65	0.348557
$364$	0	0
$365$	5427.52	0.778327
$366$	0	0
$367$	5582.93	0.794078	0.397039	−	0.917802i	$-0.370038\pi$
$367$	5582.93	0.794078	0.397039	+	0.917802i	$0.370038\pi$
$368$	0	0
$369$	944.519	0.133251
$370$	0	0
$371$	2964.80	0.414892
$372$	0	0
$373$	7140.03	0.991144	0.495572	−	0.868567i	$-0.334959\pi$
$373$	7140.03	0.991144	0.495572	+	0.868567i	$0.334959\pi$
$374$	0	0
$375$	2977.77	0.410057
$376$	0	0
$377$	−3178.09	−0.434165
$378$	0	0
$379$	−7790.88	−1.05591	−0.527956	−	0.849272i	$-0.677041\pi$
$379$	−7790.88	−1.05591	−0.527956	+	0.849272i	$0.677041\pi$
$380$	0	0
$381$	5123.51	0.688937
$382$	0	0
$383$	−801.842	−0.106977	−0.0534885	−	0.998568i	$-0.517034\pi$
$383$	−801.842	−0.106977	−0.0534885	+	0.998568i	$0.517034\pi$
$384$	0	0
$385$	−624.600	−0.0826820
$386$	0	0
$387$	−1533.82	−0.201469
$388$	0	0
$389$	6662.75	0.868419	0.434209	−	0.900812i	$-0.357028\pi$
$389$	6662.75	0.868419	0.434209	+	0.900812i	$0.357028\pi$
$390$	0	0
$391$	1751.10	0.226488
$392$	0	0
$393$	3194.03	0.409968
$394$	0	0
$395$	−616.967	−0.0785898
$396$	0	0
$397$	−12338.5	−1.55982	−0.779912	−	0.625889i	$-0.784736\pi$
$397$	−12338.5	−1.55982	−0.779912	+	0.625889i	$0.784736\pi$
$398$	0	0
$399$	1177.35	0.147722
$400$	0	0
$401$	−14029.8	−1.74717	−0.873584	−	0.486673i	$-0.838210\pi$
$401$	−14029.8	−1.74717	−0.873584	+	0.486673i	$0.838210\pi$
$402$	0	0
$403$	9311.37	1.15095
$404$	0	0
$405$	3450.47	0.423346
$406$	0	0
$407$	721.578	0.0878804
$408$	0	0
$409$	−4300.07	−0.519865	−0.259933	−	0.965627i	$-0.583700\pi$
$409$	−4300.07	−0.519865	−0.259933	+	0.965627i	$0.583700\pi$
$410$	0	0
$411$	−2135.01	−0.256234
$412$	0	0
$413$	3991.16	0.475525
$414$	0	0
$415$	2047.29	0.242163
$416$	0	0
$417$	−5011.17	−0.588485
$418$	0	0
$419$	7487.78	0.873036	0.436518	−	0.899696i	$-0.356212\pi$
$419$	7487.78	0.873036	0.436518	+	0.899696i	$0.356212\pi$
$420$	0	0
$421$	−2175.05	−0.251794	−0.125897	−	0.992043i	$-0.540181\pi$
$421$	−2175.05	−0.251794	−0.125897	+	0.992043i	$0.540181\pi$
$422$	0	0
$423$	4832.97	0.555525
$424$	0	0
$425$	−5031.34	−0.574249
$426$	0	0
$427$	2707.75	0.306879
$428$	0	0
$429$	820.325	0.0923209
$430$	0	0
$431$	13843.3	1.54712	0.773561	−	0.633722i	$-0.218474\pi$
$431$	13843.3	1.54712	0.773561	+	0.633722i	$0.218474\pi$
$432$	0	0
$433$	−2078.52	−0.230687	−0.115344	−	0.993326i	$-0.536797\pi$
$433$	−2078.52	−0.230687	−0.115344	+	0.993326i	$0.536797\pi$
$434$	0	0
$435$	−1369.95	−0.150998
$436$	0	0
$437$	1725.58	0.188891
$438$	0	0
$439$	6746.63	0.733483	0.366741	−	0.930323i	$-0.380473\pi$
$439$	6746.63	0.733483	0.366741	+	0.930323i	$0.380473\pi$
$440$	0	0
$441$	−1128.82	−0.121889
$442$	0	0
$443$	−15909.0	−1.70623	−0.853114	−	0.521725i	$-0.825289\pi$
$443$	−15909.0	−1.70623	−0.853114	+	0.521725i	$0.825289\pi$
$444$	0	0
$445$	−4762.22	−0.507306
$446$	0	0
$447$	−701.708	−0.0742498
$448$	0	0
$449$	−10324.7	−1.08519	−0.542597	−	0.839993i	$-0.682559\pi$
$449$	−10324.7	−1.08519	−0.542597	+	0.839993i	$0.682559\pi$
$450$	0	0
$451$	−449.236	−0.0469039
$452$	0	0
$453$	5238.47	0.543322
$454$	0	0
$455$	2143.87	0.220893
$456$	0	0
$457$	5523.43	0.565372	0.282686	−	0.959212i	$-0.408775\pi$
$457$	5523.43	0.565372	0.282686	+	0.959212i	$0.408775\pi$
$458$	0	0
$459$	8540.35	0.868474
$460$	0	0
$461$	18625.4	1.88172	0.940858	−	0.338801i	$-0.110021\pi$
$461$	18625.4	1.88172	0.940858	+	0.338801i	$0.110021\pi$
$462$	0	0
$463$	−263.430	−0.0264420	−0.0132210	−	0.999913i	$-0.504208\pi$
$463$	−263.430	−0.0264420	−0.0132210	+	0.999913i	$0.504208\pi$
$464$	0	0
$465$	4013.75	0.400287
$466$	0	0
$467$	14360.5	1.42297	0.711483	−	0.702703i	$-0.248024\pi$
$467$	14360.5	1.42297	0.711483	+	0.702703i	$0.248024\pi$
$468$	0	0
$469$	−7579.27	−0.746222
$470$	0	0
$471$	4466.42	0.436946
$472$	0	0
$473$	729.520	0.0709162
$474$	0	0
$475$	−4958.00	−0.478923
$476$	0	0
$477$	9757.19	0.936585
$478$	0	0
$479$	3775.31	0.360121	0.180061	−	0.983656i	$-0.442371\pi$
$479$	3775.31	0.360121	0.180061	+	0.983656i	$0.442371\pi$
$480$	0	0
$481$	−2476.74	−0.234780
$482$	0	0
$483$	284.606	0.0268117
$484$	0	0
$485$	−5969.58	−0.558896
$486$	0	0
$487$	−17266.2	−1.60659	−0.803293	−	0.595584i	$-0.796921\pi$
$487$	−17266.2	−1.60659	−0.803293	+	0.595584i	$0.796921\pi$
$488$	0	0
$489$	4801.31	0.444014
$490$	0	0
$491$	−4162.62	−0.382600	−0.191300	−	0.981532i	$-0.561270\pi$
$491$	−4162.62	−0.382600	−0.191300	+	0.981532i	$0.561270\pi$
$492$	0	0
$493$	7245.27	0.661888
$494$	0	0
$495$	−2055.56	−0.186648
$496$	0	0
$497$	6214.22	0.560857
$498$	0	0
$499$	1003.21	0.0899993	0.0449996	−	0.998987i	$-0.485671\pi$
$499$	1003.21	0.0899993	0.0449996	+	0.998987i	$0.485671\pi$
$500$	0	0
$501$	3474.58	0.309846
$502$	0	0
$503$	−9933.52	−0.880544	−0.440272	−	0.897864i	$-0.645118\pi$
$503$	−9933.52	−0.880544	−0.440272	+	0.897864i	$0.645118\pi$
$504$	0	0
$505$	−5433.46	−0.478783
$506$	0	0
$507$	1557.93	0.136470
$508$	0	0
$509$	12355.6	1.07594	0.537971	−	0.842964i	$-0.319191\pi$
$509$	12355.6	1.07594	0.537971	+	0.842964i	$0.319191\pi$
$510$	0	0
$511$	−4665.36	−0.403882
$512$	0	0
$513$	8415.86	0.724307
$514$	0	0
$515$	−8990.72	−0.769279
$516$	0	0
$517$	−2298.67	−0.195543
$518$	0	0
$519$	−5728.67	−0.484510
$520$	0	0
$521$	17552.8	1.47601	0.738004	−	0.674796i	$-0.235768\pi$
$521$	17552.8	1.47601	0.738004	+	0.674796i	$0.235768\pi$
$522$	0	0
$523$	11067.9	0.925363	0.462681	−	0.886525i	$-0.346887\pi$
$523$	11067.9	0.925363	0.462681	+	0.886525i	$0.346887\pi$
$524$	0	0
$525$	−817.743	−0.0679794
$526$	0	0
$527$	−21227.6	−1.75463
$528$	0	0
$529$	−11749.9	−0.965716
$530$	0	0
$531$	13134.9	1.07346
$532$	0	0
$533$	1541.95	0.125308
$534$	0	0
$535$	−10237.2	−0.827276
$536$	0	0
$537$	−3194.97	−0.256747
$538$	0	0
$539$	536.891	0.0429045
$540$	0	0
$541$	21624.8	1.71853	0.859265	−	0.511531i	$-0.170921\pi$
$541$	21624.8	1.71853	0.859265	+	0.511531i	$0.170921\pi$
$542$	0	0
$543$	−5277.23	−0.417067
$544$	0	0
$545$	8746.21	0.687424
$546$	0	0
$547$	11629.0	0.908991	0.454496	−	0.890749i	$-0.349819\pi$
$547$	11629.0	0.908991	0.454496	+	0.890749i	$0.349819\pi$
$548$	0	0
$549$	8911.22	0.692753
$550$	0	0
$551$	7139.66	0.552014
$552$	0	0
$553$	530.330	0.0407810
$554$	0	0
$555$	−1067.62	−0.0816540
$556$	0	0
$557$	5890.08	0.448062	0.224031	−	0.974582i	$-0.428078\pi$
$557$	5890.08	0.448062	0.224031	+	0.974582i	$0.428078\pi$
$558$	0	0
$559$	−2503.99	−0.189459
$560$	0	0
$561$	−1870.14	−0.140744
$562$	0	0
$563$	−8752.84	−0.655219	−0.327610	−	0.944813i	$-0.606243\pi$
$563$	−8752.84	−0.655219	−0.327610	+	0.944813i	$0.606243\pi$
$564$	0	0
$565$	11088.7	0.825673
$566$	0	0
$567$	−2965.94	−0.219679
$568$	0	0
$569$	17318.1	1.27594	0.637971	−	0.770061i	$-0.279774\pi$
$569$	17318.1	1.27594	0.637971	+	0.770061i	$0.279774\pi$
$570$	0	0
$571$	−10821.9	−0.793141	−0.396571	−	0.918004i	$-0.629800\pi$
$571$	−10821.9	−0.793141	−0.396571	+	0.918004i	$0.629800\pi$
$572$	0	0
$573$	8769.08	0.639326
$574$	0	0
$575$	−1198.52	−0.0869248
$576$	0	0
$577$	−20717.4	−1.49476	−0.747381	−	0.664395i	$-0.768689\pi$
$577$	−20717.4	−1.49476	−0.747381	+	0.664395i	$0.768689\pi$
$578$	0	0
$579$	9330.45	0.669707
$580$	0	0
$581$	−1759.80	−0.125661
$582$	0	0
$583$	−4640.75	−0.329674
$584$	0	0
$585$	7055.49	0.498647
$586$	0	0
$587$	17371.7	1.22148	0.610738	−	0.791833i	$-0.290873\pi$
$587$	17371.7	1.22148	0.610738	+	0.791833i	$0.290873\pi$
$588$	0	0
$589$	−20918.2	−1.46336
$590$	0	0
$591$	9463.31	0.658661
$592$	0	0
$593$	5462.73	0.378292	0.189146	−	0.981949i	$-0.439428\pi$
$593$	5462.73	0.378292	0.189146	+	0.981949i	$0.439428\pi$
$594$	0	0
$595$	−4887.50	−0.336753
$596$	0	0
$597$	5894.57	0.404102
$598$	0	0
$599$	−25312.4	−1.72660	−0.863302	−	0.504688i	$-0.831608\pi$
$599$	−25312.4	−1.72660	−0.863302	+	0.504688i	$0.831608\pi$
$600$	0	0
$601$	2507.88	0.170214	0.0851070	−	0.996372i	$-0.472877\pi$
$601$	2507.88	0.170214	0.0851070	+	0.996372i	$0.472877\pi$
$602$	0	0
$603$	−24943.4	−1.68454
$604$	0	0
$605$	−9861.39	−0.662682
$606$	0	0
$607$	27507.4	1.83936	0.919680	−	0.392668i	$-0.128448\pi$
$607$	27507.4	1.83936	0.919680	+	0.392668i	$0.128448\pi$
$608$	0	0
$609$	1177.57	0.0783541
$610$	0	0
$611$	7889.94	0.522410
$612$	0	0
$613$	14061.2	0.926474	0.463237	−	0.886234i	$-0.346688\pi$
$613$	14061.2	0.926474	0.463237	+	0.886234i	$0.346688\pi$
$614$	0	0
$615$	664.672	0.0435808
$616$	0	0
$617$	207.463	0.0135367	0.00676837	−	0.999977i	$-0.497846\pi$
$617$	207.463	0.0135367	0.00676837	+	0.999977i	$0.497846\pi$
$618$	0	0
$619$	−8688.07	−0.564141	−0.282070	−	0.959394i	$-0.591021\pi$
$619$	−8688.07	−0.564141	−0.282070	+	0.959394i	$0.591021\pi$
$620$	0	0
$621$	2034.41	0.131462
$622$	0	0
$623$	4093.49	0.263246
$624$	0	0
$625$	−4846.04	−0.310147
$626$	0	0
$627$	−1842.88	−0.117380
$628$	0	0
$629$	5646.35	0.357925
$630$	0	0
$631$	1227.95	0.0774704	0.0387352	−	0.999250i	$-0.487667\pi$
$631$	1227.95	0.0774704	0.0387352	+	0.999250i	$0.487667\pi$
$632$	0	0
$633$	−10326.7	−0.648420
$634$	0	0
$635$	−20959.0	−1.30982
$636$	0	0
$637$	−1842.82	−0.114623
$638$	0	0
$639$	20451.0	1.26609
$640$	0	0
$641$	23606.2	1.45459	0.727293	−	0.686327i	$-0.240778\pi$
$641$	23606.2	1.45459	0.727293	+	0.686327i	$0.240778\pi$
$642$	0	0
$643$	−15280.9	−0.937200	−0.468600	−	0.883411i	$-0.655241\pi$
$643$	−15280.9	−0.937200	−0.468600	+	0.883411i	$0.655241\pi$
$644$	0	0
$645$	−1079.37	−0.0658918
$646$	0	0
$647$	−1606.56	−0.0976206	−0.0488103	−	0.998808i	$-0.515543\pi$
$647$	−1606.56	−0.0976206	−0.0488103	+	0.998808i	$0.515543\pi$
$648$	0	0
$649$	−6247.28	−0.377854
$650$	0	0
$651$	−3450.13	−0.207713
$652$	0	0
$653$	2260.12	0.135445	0.0677224	−	0.997704i	$-0.478427\pi$
$653$	2260.12	0.135445	0.0677224	+	0.997704i	$0.478427\pi$
$654$	0	0
$655$	−13066.0	−0.779438
$656$	0	0
$657$	−15353.7	−0.911730
$658$	0	0
$659$	−3218.06	−0.190224	−0.0951121	−	0.995467i	$-0.530321\pi$
$659$	−3218.06	−0.190224	−0.0951121	+	0.995467i	$0.530321\pi$
$660$	0	0
$661$	28338.5	1.66753	0.833766	−	0.552118i	$-0.186180\pi$
$661$	28338.5	1.66753	0.833766	+	0.552118i	$0.186180\pi$
$662$	0	0
$663$	6419.05	0.376011
$664$	0	0
$665$	−4816.25	−0.280852
$666$	0	0
$667$	1725.90	0.100191
$668$	0	0
$669$	9476.56	0.547661
$670$	0	0
$671$	−4238.39	−0.243847
$672$	0	0
$673$	4561.43	0.261263	0.130632	−	0.991431i	$-0.458300\pi$
$673$	4561.43	0.261263	0.130632	+	0.991431i	$0.458300\pi$
$674$	0	0
$675$	−5845.35	−0.333315
$676$	0	0
$677$	12311.0	0.698890	0.349445	−	0.936957i	$-0.386370\pi$
$677$	12311.0	0.698890	0.349445	+	0.936957i	$0.386370\pi$
$678$	0	0
$679$	5131.30	0.290017
$680$	0	0
$681$	10722.0	0.603330
$682$	0	0
$683$	−17838.7	−0.999383	−0.499691	−	0.866203i	$-0.666553\pi$
$683$	−17838.7	−0.999383	−0.499691	+	0.866203i	$0.666553\pi$
$684$	0	0
$685$	8733.82	0.487157
$686$	0	0
$687$	−6782.63	−0.376672
$688$	0	0
$689$	15928.8	0.880755
$690$	0	0
$691$	27733.1	1.52680	0.763398	−	0.645929i	$-0.223530\pi$
$691$	27733.1	1.52680	0.763398	+	0.645929i	$0.223530\pi$
$692$	0	0
$693$	1766.91	0.0968535
$694$	0	0
$695$	20499.5	1.11884
$696$	0	0
$697$	−3515.27	−0.191033
$698$	0	0
$699$	−5196.08	−0.281164
$700$	0	0
$701$	−18211.8	−0.981239	−0.490620	−	0.871374i	$-0.663230\pi$
$701$	−18211.8	−0.981239	−0.490620	+	0.871374i	$0.663230\pi$
$702$	0	0
$703$	5564.04	0.298509
$704$	0	0
$705$	3401.03	0.181688
$706$	0	0
$707$	4670.47	0.248446
$708$	0	0
$709$	−25578.7	−1.35491	−0.677454	−	0.735565i	$-0.736917\pi$
$709$	−25578.7	−1.35491	−0.677454	+	0.735565i	$0.736917\pi$
$710$	0	0
$711$	1745.32	0.0920599
$712$	0	0
$713$	−5056.66	−0.265601
$714$	0	0
$715$	−3355.76	−0.175522
$716$	0	0
$717$	7932.52	0.413173
$718$	0	0
$719$	6591.90	0.341914	0.170957	−	0.985278i	$-0.445314\pi$
$719$	6591.90	0.341914	0.170957	+	0.985278i	$0.445314\pi$
$720$	0	0
$721$	7728.21	0.399186
$722$	0	0
$723$	−1517.71	−0.0780693
$724$	0	0
$725$	−4958.94	−0.254028
$726$	0	0
$727$	10869.3	0.554496	0.277248	−	0.960798i	$-0.410578\pi$
$727$	10869.3	0.554496	0.277248	+	0.960798i	$0.410578\pi$
$728$	0	0
$729$	−4406.97	−0.223897
$730$	0	0
$731$	5708.50	0.288832
$732$	0	0
$733$	33799.2	1.70314	0.851569	−	0.524242i	$-0.175651\pi$
$733$	33799.2	1.70314	0.851569	+	0.524242i	$0.175651\pi$
$734$	0	0
$735$	−794.364	−0.0398647
$736$	0	0
$737$	11863.7	0.592951
$738$	0	0
$739$	9418.06	0.468808	0.234404	−	0.972139i	$-0.424686\pi$
$739$	9418.06	0.468808	0.234404	+	0.972139i	$0.424686\pi$
$740$	0	0
$741$	6325.48	0.313593
$742$	0	0
$743$	10280.3	0.507601	0.253800	−	0.967257i	$-0.418319\pi$
$743$	10280.3	0.507601	0.253800	+	0.967257i	$0.418319\pi$
$744$	0	0
$745$	2870.52	0.141165
$746$	0	0
$747$	−5791.53	−0.283669
$748$	0	0
$749$	8799.65	0.429282
$750$	0	0
$751$	−29872.5	−1.45148	−0.725740	−	0.687969i	$-0.758502\pi$
$751$	−29872.5	−1.45148	−0.725740	+	0.687969i	$0.758502\pi$
$752$	0	0
$753$	11182.6	0.541193
$754$	0	0
$755$	−21429.3	−1.03297
$756$	0	0
$757$	−21602.8	−1.03721	−0.518603	−	0.855015i	$-0.673548\pi$
$757$	−21602.8	−1.03721	−0.518603	+	0.855015i	$0.673548\pi$
$758$	0	0
$759$	−445.489	−0.0213046
$760$	0	0
$761$	−23523.6	−1.12054	−0.560269	−	0.828311i	$-0.689302\pi$
$761$	−23523.6	−1.12054	−0.560269	+	0.828311i	$0.689302\pi$
$762$	0	0
$763$	−7518.03	−0.356711
$764$	0	0
$765$	−16084.8	−0.760192
$766$	0	0
$767$	21443.1	1.00947
$768$	0	0
$769$	41085.3	1.92662	0.963312	−	0.268385i	$-0.0864898\pi$
$769$	41085.3	1.92662	0.963312	+	0.268385i	$0.0864898\pi$
$770$	0	0
$771$	−2527.24	−0.118050
$772$	0	0
$773$	−2448.60	−0.113933	−0.0569664	−	0.998376i	$-0.518143\pi$
$773$	−2448.60	−0.113933	−0.0569664	+	0.998376i	$0.518143\pi$
$774$	0	0
$775$	14529.0	0.673416
$776$	0	0
$777$	917.700	0.0423711
$778$	0	0
$779$	−3464.03	−0.159322
$780$	0	0
$781$	−9727.00	−0.445658
$782$	0	0
$783$	8417.47	0.384184
$784$	0	0
$785$	−18271.1	−0.830729
$786$	0	0
$787$	5101.92	0.231085	0.115543	−	0.993303i	$-0.463139\pi$
$787$	5101.92	0.231085	0.115543	+	0.993303i	$0.463139\pi$
$788$	0	0
$789$	−6019.11	−0.271592
$790$	0	0
$791$	−9531.58	−0.428450
$792$	0	0
$793$	14547.8	0.651458
$794$	0	0
$795$	6866.28	0.306317
$796$	0	0
$797$	3696.45	0.164285	0.0821423	−	0.996621i	$-0.473824\pi$
$797$	3696.45	0.164285	0.0821423	+	0.996621i	$0.473824\pi$
$798$	0	0
$799$	−17987.1	−0.796419
$800$	0	0
$801$	13471.7	0.594256
$802$	0	0
$803$	7302.60	0.320926
$804$	0	0
$805$	−1164.26	−0.0509747
$806$	0	0
$807$	14477.3	0.631507
$808$	0	0
$809$	−1428.72	−0.0620903	−0.0310452	−	0.999518i	$-0.509884\pi$
$809$	−1428.72	−0.0620903	−0.0310452	+	0.999518i	$0.509884\pi$
$810$	0	0
$811$	−33554.6	−1.45285	−0.726426	−	0.687245i	$-0.758820\pi$
$811$	−33554.6	−1.45285	−0.726426	+	0.687245i	$0.758820\pi$
$812$	0	0
$813$	−15464.8	−0.667126
$814$	0	0
$815$	−19641.0	−0.844166
$816$	0	0
$817$	5625.28	0.240886
$818$	0	0
$819$	−6064.72	−0.258753
$820$	0	0
$821$	41277.3	1.75467	0.877336	−	0.479876i	$-0.159318\pi$
$821$	41277.3	1.75467	0.877336	+	0.479876i	$0.159318\pi$
$822$	0	0
$823$	−30683.3	−1.29958	−0.649788	−	0.760115i	$-0.725142\pi$
$823$	−30683.3	−1.29958	−0.649788	+	0.760115i	$0.725142\pi$
$824$	0	0
$825$	1280.00	0.0540167
$826$	0	0
$827$	17665.3	0.742786	0.371393	−	0.928476i	$-0.378880\pi$
$827$	17665.3	0.742786	0.371393	+	0.928476i	$0.378880\pi$
$828$	0	0
$829$	−29345.1	−1.22943	−0.614716	−	0.788749i	$-0.710729\pi$
$829$	−29345.1	−1.22943	−0.614716	+	0.788749i	$0.710729\pi$
$830$	0	0
$831$	2680.88	0.111912
$832$	0	0
$833$	4201.17	0.174744
$834$	0	0
$835$	−14213.7	−0.589083
$836$	0	0
$837$	−24662.0	−1.01845
$838$	0	0
$839$	8253.41	0.339618	0.169809	−	0.985477i	$-0.445685\pi$
$839$	8253.41	0.339618	0.169809	+	0.985477i	$0.445685\pi$
$840$	0	0
$841$	−17248.0	−0.707203
$842$	0	0
$843$	−16498.7	−0.674075
$844$	0	0
$845$	−6373.14	−0.259459
$846$	0	0
$847$	8476.61	0.343872
$848$	0	0
$849$	890.989	0.0360173
$850$	0	0
$851$	1345.02	0.0541796
$852$	0	0
$853$	−30912.0	−1.24081	−0.620403	−	0.784284i	$-0.713031\pi$
$853$	−30912.0	−1.24081	−0.620403	+	0.784284i	$0.713031\pi$
$854$	0	0
$855$	−15850.3	−0.633999
$856$	0	0
$857$	−44577.2	−1.77681	−0.888407	−	0.459056i	$-0.848188\pi$
$857$	−44577.2	−1.77681	−0.888407	+	0.459056i	$0.848188\pi$
$858$	0	0
$859$	−13942.0	−0.553776	−0.276888	−	0.960902i	$-0.589303\pi$
$859$	−13942.0	−0.553776	−0.276888	+	0.960902i	$0.589303\pi$
$860$	0	0
$861$	−571.336	−0.0226145
$862$	0	0
$863$	8294.88	0.327185	0.163593	−	0.986528i	$-0.447692\pi$
$863$	8294.88	0.327185	0.163593	+	0.986528i	$0.447692\pi$
$864$	0	0
$865$	23434.6	0.921157
$866$	0	0
$867$	−4853.46	−0.190118
$868$	0	0
$869$	−830.115	−0.0324047
$870$	0	0
$871$	−40720.8	−1.58412
$872$	0	0
$873$	16887.1	0.654689
$874$	0	0
$875$	10470.8	0.404546
$876$	0	0
$877$	15267.3	0.587843	0.293922	−	0.955830i	$-0.405040\pi$
$877$	15267.3	0.587843	0.293922	+	0.955830i	$0.405040\pi$
$878$	0	0
$879$	−15654.9	−0.600712
$880$	0	0
$881$	14667.3	0.560902	0.280451	−	0.959868i	$-0.409516\pi$
$881$	14667.3	0.560902	0.280451	+	0.959868i	$0.409516\pi$
$882$	0	0
$883$	−25709.9	−0.979851	−0.489926	−	0.871764i	$-0.662976\pi$
$883$	−25709.9	−0.979851	−0.489926	+	0.871764i	$0.662976\pi$
$884$	0	0
$885$	9243.24	0.351083
$886$	0	0
$887$	12993.4	0.491856	0.245928	−	0.969288i	$-0.420907\pi$
$887$	12993.4	0.491856	0.245928	+	0.969288i	$0.420907\pi$
$888$	0	0
$889$	18015.9	0.679678
$890$	0	0
$891$	4642.53	0.174557
$892$	0	0
$893$	−17724.9	−0.664213
$894$	0	0
$895$	13069.9	0.488131
$896$	0	0
$897$	1529.09	0.0569172
$898$	0	0
$899$	−20922.2	−0.776190
$900$	0	0
$901$	−36313.9	−1.34272
$902$	0	0
$903$	927.801	0.0341919
$904$	0	0
$905$	21587.9	0.792935
$906$	0	0
$907$	18009.5	0.659311	0.329656	−	0.944101i	$-0.393067\pi$
$907$	18009.5	0.659311	0.329656	+	0.944101i	$0.393067\pi$
$908$	0	0
$909$	15370.5	0.560845
$910$	0	0
$911$	−30563.6	−1.11154	−0.555772	−	0.831335i	$-0.687577\pi$
$911$	−30563.6	−1.11154	−0.555772	+	0.831335i	$0.687577\pi$
$912$	0	0
$913$	2754.59	0.0998506
$914$	0	0
$915$	6270.96	0.226570
$916$	0	0
$917$	11231.2	0.404458
$918$	0	0
$919$	−45270.6	−1.62496	−0.812480	−	0.582989i	$-0.801883\pi$
$919$	−45270.6	−1.62496	−0.812480	+	0.582989i	$0.801883\pi$
$920$	0	0
$921$	−4466.93	−0.159816
$922$	0	0
$923$	33386.8	1.19062
$924$	0	0
$925$	−3864.58	−0.137369
$926$	0	0
$927$	25433.6	0.901131
$928$	0	0
$929$	19494.9	0.688491	0.344245	−	0.938880i	$-0.388135\pi$
$929$	19494.9	0.688491	0.344245	+	0.938880i	$0.388135\pi$
$930$	0	0
$931$	4139.93	0.145737
$932$	0	0
$933$	1581.59	0.0554973
$934$	0	0
$935$	7650.30	0.267585
$936$	0	0
$937$	−43948.7	−1.53227	−0.766137	−	0.642678i	$-0.777823\pi$
$937$	−43948.7	−1.53227	−0.766137	+	0.642678i	$0.777823\pi$
$938$	0	0
$939$	−19342.5	−0.672225
$940$	0	0
$941$	−5788.21	−0.200521	−0.100260	−	0.994961i	$-0.531968\pi$
$941$	−5788.21	−0.200521	−0.100260	+	0.994961i	$0.531968\pi$
$942$	0	0
$943$	−837.376	−0.0289170
$944$	0	0
$945$	−5678.23	−0.195463
$946$	0	0
$947$	17685.9	0.606878	0.303439	−	0.952851i	$-0.401865\pi$
$947$	17685.9	0.606878	0.303439	+	0.952851i	$0.401865\pi$
$948$	0	0
$949$	−25065.4	−0.857382
$950$	0	0
$951$	−17052.1	−0.581441
$952$	0	0
$953$	24689.0	0.839197	0.419598	−	0.907710i	$-0.362171\pi$
$953$	24689.0	0.839197	0.419598	+	0.907710i	$0.362171\pi$
$954$	0	0
$955$	−35872.2	−1.21550
$956$	0	0
$957$	−1843.23	−0.0622604
$958$	0	0
$959$	−7507.38	−0.252790
$960$	0	0
$961$	31508.2	1.05764
$962$	0	0
$963$	28959.7	0.969069
$964$	0	0
$965$	−38168.7	−1.27326
$966$	0	0
$967$	−28733.8	−0.955549	−0.477775	−	0.878482i	$-0.658556\pi$
$967$	−28733.8	−0.955549	−0.477775	+	0.878482i	$0.658556\pi$
$968$	0	0
$969$	−14420.5	−0.478075
$970$	0	0
$971$	−45163.7	−1.49266	−0.746330	−	0.665576i	$-0.768186\pi$
$971$	−45163.7	−1.49266	−0.746330	+	0.665576i	$0.768186\pi$
$972$	0	0
$973$	−17620.9	−0.580575
$974$	0	0
$975$	−4393.44	−0.144310
$976$	0	0
$977$	4887.81	0.160056	0.0800282	−	0.996793i	$-0.474499\pi$
$977$	4887.81	0.160056	0.0800282	+	0.996793i	$0.474499\pi$
$978$	0	0
$979$	−6407.46	−0.209176
$980$	0	0
$981$	−24741.9	−0.805247
$982$	0	0
$983$	49892.9	1.61886	0.809429	−	0.587218i	$-0.199777\pi$
$983$	49892.9	1.61886	0.809429	+	0.587218i	$0.199777\pi$
$984$	0	0
$985$	−38712.2	−1.25226
$986$	0	0
$987$	−2923.44	−0.0942799
$988$	0	0
$989$	1359.83	0.0437209
$990$	0	0
$991$	−21254.4	−0.681300	−0.340650	−	0.940190i	$-0.610647\pi$
$991$	−21254.4	−0.681300	−0.340650	+	0.940190i	$0.610647\pi$
$992$	0	0
$993$	−14463.5	−0.462221
$994$	0	0
$995$	−24113.3	−0.768284
$996$	0	0
$997$	14119.7	0.448520	0.224260	−	0.974529i	$-0.428004\pi$
$997$	14119.7	0.448520	0.224260	+	0.974529i	$0.428004\pi$
$998$	0	0
$999$	6559.86	0.207753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.b.1.7	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.b.1.7	✓	15	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q-1.99072 q^{3} +8.14355 q^{5} -7.00000 q^{7} -23.0370 q^{9} +O(q^{10})\)	\(q-1.99072 q^{3} +8.14355 q^{5} -7.00000 q^{7} -23.0370 q^{9} +10.9570 q^{11} -37.6085 q^{13} -16.2115 q^{15} +85.7382 q^{17} +84.4884 q^{19} +13.9350 q^{21} +20.4238 q^{23} -58.6825 q^{25} +99.6096 q^{27} +84.5046 q^{29} -247.587 q^{31} -21.8122 q^{33} -57.0049 q^{35} +65.8557 q^{37} +74.8679 q^{39} -41.0000 q^{41} +66.5805 q^{43} -187.603 q^{45} -209.791 q^{47} +49.0000 q^{49} -170.681 q^{51} -423.543 q^{53} +89.2286 q^{55} -168.193 q^{57} -570.165 q^{59} -386.821 q^{61} +161.259 q^{63} -306.267 q^{65} +1082.75 q^{67} -40.6580 q^{69} -887.746 q^{71} +666.481 q^{73} +116.820 q^{75} -76.6987 q^{77} -75.7614 q^{79} +423.706 q^{81} +251.401 q^{83} +698.214 q^{85} -168.225 q^{87} -584.785 q^{89} +263.260 q^{91} +492.875 q^{93} +688.036 q^{95} -733.043 q^{97} -252.416 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) \) 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9 - 20 * q^11 - 70 * q^13 - 20 * q^15 + 160 * q^17 - 6 * q^19 - 118 * q^23 + 569 * q^25 + 18 * q^27 - 162 * q^29 - 164 * q^31 - 292 * q^33 - 42 * q^35 - 410 * q^37 - 206 * q^39 - 615 * q^41 - 1022 * q^43 + 196 * q^45 - 628 * q^47 + 735 * q^49 - 1994 * q^51 - 512 * q^53 - 1128 * q^55 - 266 * q^57 - 144 * q^59 - 256 * q^61 - 623 * q^63 - 1000 * q^65 - 2670 * q^67 + 108 * q^69 - 1048 * q^71 - 606 * q^73 - 3796 * q^75 + 140 * q^77 - 1386 * q^79 - 2541 * q^81 - 2022 * q^83 - 2848 * q^85 - 3700 * q^87 - 500 * q^89 + 490 * q^91 - 2194 * q^93 - 5230 * q^95 + 1326 * q^97 - 2732 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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